Finite Group Databasebeta

← #10606129

Subgroups — #10606129

Order 640.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D5C211yesmaximal, characteristic
2C2×#4033C211yesmaximal, characteristic, complemented
3C2×#4031C211yesmaximal, characteristic
4C5×#385C211yesmaximal, characteristic, complemented
5C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
6#65279C211yesmaximal, characteristic, complemented
7#65278C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
9#65246C211yesmaximal, characteristic
10#65240C211yesmaximal, characteristic, complemented
11#65233C211yesmaximal, characteristic, complemented
12#64959C211yesmaximal, characteristic, complemented
13#64955C211yesmaximal, characteristic, complemented
14#64938C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C23×C4×C5C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C4×C22⋊D5—24no—
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44#4033—24no—
45#4031C418yescomplemented
46C5×#95C2211yescharacteristic
47C5×#95C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#2367—51nomaximal, Sylow
55C24×C5C2311yescharacteristic
56C23×D5C2311yescharacteristic
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—41no—
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C2×C22⋊D5—22no—
67C2×C22⋊D5—22no—
68C2×C22⋊D5C2×C414yes—
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—41no—
79C2×Dic10—22no—
80C2×Dic10—22no—
81C2×Dic10C2×C414yescomplemented
82C5×C4⋊C4—22no—
83C5×C4⋊C4—22no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4C2×C414yescomplemented
86C22⋊Dic5—22no—
87C22⋊Dic5—22no—
88C22⋊Dic5C2×C414yescomplemented
89#685—22no—
90#685—22no—
91#685—41no—
92#685C2×C414yescomplemented
93#685C2×C414yescomplemented
94C4⋊Dic5—22no—
95#683—22no—
96#683—22no—
97#683D414yes—
98C4×Dic5—22no—
99C4×Dic5—22no—
100C4×Dic5—22no—
101C4×Dic5—22no—
102C4×Dic5—41no—
103C4×Dic5—41no—
104C4×Dic5C2×C414yescomplemented
105C2×#124—51no—
106C2×C4×Q8—51no—
107C2×C4×D4—51no—
108#391—51no—
109#390—51no—
110#388—51no—
111#388—51no—
112#388—51no—
113#388—51no—
114#385—51no—
115#385D511yescharacteristic, complemented
116C4×C4⋊C4—51no—
117C4×C4⋊C4—51no—
118C4×C22⋊C4—51no—
119C4×C22⋊C4—51no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5C22×C412yes—
127C23×C5C2411yescharacteristic
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5C22×C412yes—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—44no—
150C2×C4×C5C22×C412yes—
151C2×C4×C5C22×C412yes—
152C2×C4×C5C2×D412yes—
153C2×C4×C5#4112yes—
154C22⋊D5—28no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5C22×C412yes—
170C2×Dic5C22×C412yes—
171C2×Dic5C22×C412yes—
172C2×Dic5C2×D412yes—
173C2×Dic5C2×D412yes—
174C2×Dic5#4112yes—
175C2×Dic5#4112yes—
176C4×D5—44no—
177Dic10—28no—
178C22×Q8—51no—
179C22×D4—51no—
180C23×C4—51no—
181C23×C4C2×D511yescharacteristic
182#124—58no—
183C4×Q8—104no—
184C4×D4—104no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4C2×D511yescharacteristic
190C2×C4⋊C4C2×D511yescharacteristic
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4C2×D511yescharacteristic
200C2×C22⋊C4C2×D511yescharacteristic
201C2×C42—51no—
202C2×C42—51no—
203C2×C42—51no—
204C2×C42—51no—
205C2×C42—51no—
206C2×C42—51no—
207C2×C42—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95C2×D511yescharacteristic
215#95C2×D511yescharacteristic
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5C23×C411yescharacteristic
224C22×C5C22×D411yescharacteristic
225C22×C5C2×#4111yescharacteristic
226C22×C5C2×#4111yescharacteristic
227C22×C5C2×#4111yescharacteristic
228C22×C5H(5,𝔽2)11yescharacteristic
229C22×C52−1+411yescharacteristic
230C2×D5—22no—
231C2×D5—22no—
232C2×D5—22no—
233C2×D5—41no—
234C2×D5—41no—
235C2×D5—41no—
236C2×D5—41no—
237C4×C5—22no—
238C4×C5—22no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—41no—
242C4×C5—41no—
243Dic5—22no—
244Dic5—22no—
245Dic5—22no—
246Dic5—22no—
247Dic5—41no—
248Dic5—41no—
249Dic5C4×D414yes—
250C24—51no—
251C24C22×D511yescharacteristic
252C2×Q8—54no—
253C2×Q8—102no—
254C2×Q8—102no—
255C2×D4—54no—
256C2×D4—102no—
257C2×D4—102no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—102no—
269C22×C4—102no—
270C22×C4—201no—
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C22×C4C22×D511yescharacteristic
276C22×C4C22×D511yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—54no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C22⋊C4—41no—
284C22⋊C4—54no—
285C22⋊C4—54no—
286C22⋊C4—54no—
287C22⋊C4—102no—
288C22⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—201no—
292C22⋊C4C4×D514yescomplemented
293C42—54no—
294C42—102no—
295C42—102no—
296C42—102no—
297C42—102no—
298C42—201no—
299C42—201no—
300C2×C5—42no—
301C2×C5C2×C4×D411yescharacteristic
302C2×C5C4×#4111yescharacteristic
303C2×C5#52011yescharacteristic
304C2×C5#54611yescharacteristic
305C2×C5#54811yescharacteristic
306C2×C5#55611yescharacteristic
307C2×C5#55911yescharacteristic
308D5—42no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—52no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23—101no—
322C23C2×C4×D512yes—
323C23C23×D511yescharacteristic
324Q8—108no—
325D4—108no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—52no—
342C2×C4—52no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—52no—
346C2×C4—52no—
347C2×C4—52no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—204no—
363C2×C4C2×C4×D512yes—
364C2×C4C2×C4×D512yes—
365C2×C4D4×D512yes—
366C2×C4#71112yes—
367C5#236711yescharacteristic, complemented, Sylow
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—102no—
376C22—102no—
377C22—102no—
378C22—201no—
379C22—201no—
380C22—201no—
381C22—201no—
382C22C22×C4×D511yescharacteristic
383C22C2×D4×D511yescharacteristic
384C22C2×#71111yescharacteristic
385C22#414411yescharacteristic
386C22#414711yescharacteristic
387C22D5×#4111yescharacteristic
388C22D5×#4111yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—54no—
396C4—102no—
397C4—102no—
398C4—102no—
399C4—102no—
400C4—201no—
401C4—201no—
402C2—42no—
403C2—202no—
404C2#6583811yescharacteristic
405C2C4×#71111yescharacteristic
406C2C4×D4×D511yescharacteristic
407C2#6592511yescharacteristic
408C2#6596711yescharacteristic
409C2#6597511yescharacteristic
410C2#6599811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#385C2
socle, cosocleC23×C5C24
prime radicalC22×C22⋊D5C22
hypercenter#385D5
hypocenterC5#2367
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#4033C2semidirect
4. C5×#385C2semidirect
5. C4×C22⋊Dic5C2semidirect
6. #65279C2semidirect
7. #65278C2semidirect
8. C4⋊C4×Dic5C2semidirect
10. #65240C2semidirect
11. #65233C2semidirect
12. #64959C2semidirect
13. #64955C2semidirect
14. #64938C2semidirect
15. #64935C2semidirect
45. #4031C4semidirect
81. C2×Dic10C2×C4semidirect
85. C5×C22⋊C4C2×C4semidirect
88. C22⋊Dic5C2×C4semidirect
92. #685C2×C4semidirect
93. #685C2×C4semidirect
104. C4×Dic5C2×C4semidirect
115. #385D5semidirect
292. C22⋊C4C4×D5semidirect
367. C5#2367semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×C22⋊D52
C2×#40332
C2×#40312
C5×#3852
C4×C22⋊Dic52
#652792
#652782
C4⋊C4×Dic52
#652462
#652402
#652332
#649592
#649552
#649382
#649352
#23675